Problem: The sum of two numbers is $129$, and their difference is $53$. What are the two numbers?
Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 129}$ ${x-y = 53}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 182 $ $ x = \dfrac{182}{2} $ ${x = 91}$ Now that you know ${x = 91}$ , plug it back into $ {x+y = 129}$ to find $y$ ${(91)}{ + y = 129}$ ${y = 38}$ You can also plug ${x = 91}$ into $ {x-y = 53}$ and get the same answer for $y$ ${(91)}{ - y = 53}$ ${y = 38}$ Therefore, the larger number is $91$, and the smaller number is $38$.